The invention relates to financial securities trading such as, e.g., trading in stocks, bonds and financial derivative instruments, including futures, options and collateralized mortgage obligations.
In financial securities trading, which includes the initial offer for sale, the value of a security may be estimated, e.g., based on expected future cash flow. Such cash flow may depend on variable interest rates, for example, and these and other relevant variables may be viewed as stochastic variables.
It is well known that the value of a financial security which depends on stochastic variables can be estimated in terms of a multi-dimensional integral. The dimension of such an integral may be very high.
In collateralized mortgage obligations (CMO), for example, instruments or securities variously called tranches, shares, participations, classes or contracts have cash flows which are determined by dividing and distributing the cash flow of an underlying collection or pool of mortgages on a monthly basis according to pre-specified rules. The present value of a tranche can be estimated on the basis of the expected monthly cash flows over the remaining term of the tranche, and an estimate of the present value of a tranche can be represented as a multi-dimensional integral whose dimension is the number of payment periods of the tranche. For a typical instrument with a 30-year term and with monthly payments, this dimension is 360.
Usually, such a high-dimensional integral can be evaluated only approximately, by numerical integration. This involves the generation of points in the domain of integration, evaluating or "sampling" the integrand at the generated points, and combining the resulting integrand values, e.g., by averaging. Well known for numerical integration in securities trading is the so-called Monte Carlo method in which points in the domain of integration are generated at random.
With integrands arising in financial securities trading, the computational work in combining the sampled values is negligible as compared with producing the integrand values. Thus, numerical integration methods in securities trading may be compared based on the number of samples required for obtaining a sufficiently accurate approximation to the integral.